What fixes the constants of an octonionic matrix model? : Invariant counting, the fermion-induced potential, two-block dynamics, and the data [Paper P3 of a ten-part series P1-P10]

Paper P3 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P3 — Symmetry, induced interactions and coefficient freedom A companion paper worked out a G₂-invariant matrix model for seven Hermitian N× N matrices with an octonionic cubic term, a commutator quartic, a mass term and Clifford-coupled fermions, as a benchmark for the spacetime--matter programme. This paper asks the question that benchmark leaves open: are the constants of such a model unique, or does the programme face a landscape? We answer it in four independent but mutually informing analyses. (A)~Counting: at polynomial degree four the symmetry leaves exactly two independent single-trace quartics for seven matrices under G₂ (the coassociative four-form, although an independent invariant tensor, is annihilated by cyclicity of the trace) and three for an E₈-adjoint multi-matrix action, so symmetry alone leaves two or three free dimensionless ratios; the programme's written action is purely kinetic and fixes none of them; and the three natural selection principles (commutator structure, square of a gradient, spectral action) are mutually incompatible, none reproducing the benchmark. A spectral action built from the model's own Clifford operator fixes the ratio of the two quartics to λ/g²=1/2 for every spectral function but cannot produce the cubic at any order, because that operator is (7)-covariant. (B)~Induction: integrating out the model's own fermions in a thermal ensemble gives a concave induced potential with negative mass, λ/g²=1/2 for a (7)-invariant fermion mass, and no cubic unless the singlet and vector parts of the Clifford module are split in mass; the induced t=μ²/(g²m²) is then negative for every mass, and the sphere condition t≥8, m²>0 is never met. With a bare mass the state selects the geometry: a commuting condensate just below Tc, the fuzzy sphere below ≈0.7\,Tc at $N=3$, and S²× S² rather than the sphere at $N=4$; at the benchmark point, in the leading semiclassical estimate at =1, the sphere is the lowest of the configurations compared only once the bosonic zero-point energy is included. (C)~Dynamics: classically, the string sector between two sphere blocks is linearly unstable along every encounter that enters the tachyonic window and secularly unstable through the orientation Goldstones even at frozen separation, so two atoms merge into a chaotic blob; the low-frequency force weight that record formation needs appears only above a chaos threshold at which the strings carry far more energy than the blocks bind. (D)~Data: every identification of the model's exact spectra with Standard-Model masses fails on a relation of the model itself. The verdict is that the programme faces neither unique constants nor a string-like landscape: the vacuum set at fixed couplings is small and discrete, but the couplings are undetermined by the action, the one internal mechanism that determines them points away from the geometric phase, and a successful identification would need a (7)-breaking ingredient and a continuous family sector that the present action does not contain. All calculations are supplied as scripts with assertions and result files.